How to save the render result within Blender
No, Blender doesn't have this functionality.

Its possible to write an addon to do this, on-save it could save the render result to a Multi-Layer-EXR image, and on load it could check for that image, loading it if its available. (probably not all that hard to get basic functionality, but you will loose render slots).



An alternative which is admittedly little clumsy, you could use a file-output node in the compositor and save this in your user-defaults,
so your renders are automatically saved. In the user default you could have an image view pointing to this file, so on-load it would read the last rendered image.